Solvable time-like cosets and holography beyond AdS

We build a novel time-like coset sigma-model describing type-II superstring theory in a charged rotating black-brane background that interpolates between a local AdS3 in the IR and a linear-dilaton geometry in the UV. This allows one to peform a systematic study of holography in non-AdS backgrounds which are smoothly connected to AdS3. We construct massless closed string states vertex operators in the NS-NS sector, calculate the corresponding two-point correlation functions, and discuss holographic interpretation of our results from 4+1 dimensional boundary field theory point of view. Compactifying the theory on 𝕋4, we show that the spectrum of a single long string with unit winding agrees with the spectrum of a CFT2 deformed by TT¯\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ T\overline{T} $$\end{document}. We also calculate correlation functions of operators of the dual 1+1 dimensional non-conformal boundary field theory using worldsheet techniques.


JHEP08(2022)244 1 Introduction
String theory on group manifolds and coset spaces supplies a rich class of models describing strings propagating on a non-trivial curved background, that are amenable to a formulation in terms of a two-dimensional world-sheet theory, exact in string length. This powerful feature follows from the potent combination of (super-)conformal and affine Kac-Moody symmetries enjoyed by these theories. The world-sheet action that respects these symmetries and describes string theory on group manifolds is given by the celebrated Wess-Zumino-Witten (WZW) model, while its generalization to group cosets is described by the gauged WZW (gWZW) model [1][2][3].
A particular class of the (g)WZW models, solvable in the limit of small string coupling g s , yet exact in the string length s , is given by the world-sheet models with the target space based on the SL(2, R) and SU(2) group manifolds. An enormous technical simplification due to such a choice of the target space group manifold is afforded by the available properties of the SL(2, R) and SU(2) primary operators, in particular, the known structure of their correlation functions in the exact world-sheet theory. This knowledge assists one immensely in a calculation of various observables exact in the WZW level k that simultaneously translates into the stringy, beyond gravity, accessibility of the large curvature regime: a notoriously coveted goal of practitioners in the field.
One of the fruitful applications of the SL(2, R) and SU(2) based (g)WZW models is found in the field of holographic correspondence. Original examples are given by the holographic (bulk) description of a two-dimensional conformal field theory (CFT) [4][5][6][7][8][9], and holographic description of Little String Theory (LST) [10][11][12][13][14][15][16][17][18][19][20][21][22][23]. The former is given by the superconformal WZW model on AdS 3 × S 3 × T 4 (where AdS 3 ∼ SL(2, R), and S 3 ∼ SU(2)), while the latter is described by the superconformal gWZW model on SL(2,R) U(1) × SU(2) × R 5 . In the case when microscopic description of coset models is provided by a stack of k coincident NS5 branes wrapping R 5 or T 4 × S 1 and p fundamental strings (F1) along R 1 or S 1 , the AdS 3 background found in the near-horizon limit of the F1 strings is described by the SL(2, R) WZW model at level k, while the coset SL(2, R)/U(1) describes the lineardilaton background, sourced by a stack of k NS5-branes in its near-horizon limit. The corresponding dual field theory interpolates between a CFT 2 , obtained by taking the decoupling limit of the F1-strings, and the world-volume theory of the NS5-branes, namely LST, obtained by taking the decoupling limit of the NS5 branes. The NS5-branes source the NS flux, equal to k, through the S 3 ∼ SU(2) sphere surrounding the NS5-branes in the ten-dimensional space-time. The radius of both AdS 3 and S 3 behaves as R = √ k s , which defines the small-curvature semi-classical limit (amenable to the supergravity approximation) at k → ∞. However, an exact solvability of the SL(2, R) × SU(2) WZW model allows one to carry out calculations exactly in k, furnishing an exact world-sheet description.
One can construct a more complicated class of cosets describing dynamics of NS5-F1 system. Coset systems, described by the gWZW model, require one to impose anomalycancellation conditions. These can be formulated as time-like or null constraints (for cosets that include the time-like direction in target space) on the gauge parameters. Null cosets have been extensively studied in [24][25][26][27] and shown to describe micro-state geometries JHEP08(2022)244 with world-sheet control. On the other hand, time-like cosets help us understand dynamics of black holes, scattering in a black hole background, and resolution of cosmological singularities [15,[28][29][30][31][32].
In [31,33] collective excitations of the LST were studied holographically, by analyzing the pole structure of the stress-energy tensor two-point function, described in terms of the graviton scattering amplitudes in the bulk. 1 The latter were calculated exactly, in terms of the two-point correlation functions of vertex operators on the world-sheet, describing massless closed string states in the Neveu-Schwarz-Neveu-Schwarz (NS-NS) sector of the supergravity multiplet. In particular, the diffusion mode was located, and its existence was independently verified by the supergravity analysis. Additionally, in [31] a finite-density version of this model was considered, by constructing the dual bulk configuration with a non-trivial gauge field profile. Such a background was explicitly demonstrated in [32] to appear in the configuration involving the near-horizon limit of a stack of NS5-branes with a large number of fundamental strings (F1).
In this paper, our goal is to further broaden our understanding of the asymptotically linear dilaton backgrounds, construct spectrum of string excitations on top of these backgrounds, calculate various correlation functions of vertex operators corresponding to massless string states, and comment on application of these results to holography. To this end, we consider type-II superstring theory on SL(2,R)×SU(2)×U(1)x U (1) × T 4 ten-dimensional coset space-time, and perform time-like gauging of the U(1) sub-group in denominator. We formulate exact world-sheet action of the superconformal gWZW model on this coset, describing type-II superstring theory in the corresponding background. We then proceed to deriving the exact in k gravity background, consisting of non-trivial configurations of the metric, dilaton, and B-field. The obtained background asymptotes to the two-dimensional linear dilaton background times S 3 × S 1 × T 4 . At the same time, the obtained geometry possesses an event horizon with non-vanishing angular velocities along two of the S 3 and S 1 directions.
Compactifying the U(1) x ∼ S 1 direction and performing the Kaluza-Klein (KK) reduction, we obtain the nine-dimensional space-time with two non-trivial U(1) gauge field profiles, non-vanishing angular velocities along two of the S 3 directions, and an event horizon. Such a configuration describes a holographically dual theory at finite temperature, characterized by finite values of chemical potentials and charge densities corresponding to two U(1) charges. We provide a complete thermodynamic description of the ten-and ninedimensional backgrounds, by calculating the corresponding temperature, entropy, angular momenta, energy, grand potential, and charges. We demonstrate that for the particular choice of the coset gauging parameters our results reproduce the known results in the literature, while largely extending those results to a more general case of a rotating charged asymptotically linear-dilaton background with a black brane event horizon.
We subsequently discuss collective excitations of the boundary theory by inspecting the structure of two-point correlation functions of the stress-energy tensor and the currents corresponding to two conserved U(1) charges. These can be determined holographically in terms of the two-point functions of the vertex operators of the world-sheet theory, describing JHEP08(2022)244 massless excitations of the NS-NS sector of the type-II supergravity background, involving the metric, dilaton, B-field, and gauge fields. We construct the corresponding vertex operators by performing the coset BRST quantization of the world-sheet theory, ensuring the U(1) symmetry at the quantum level. In particular, we determine the tachyon vertex operator, and study its asymptotic behavior. We then construct the most general massless NS-NS physical states, and calculate their two-point correlation functions.
The string theory model that we study in this paper, at zero temperature, interpolates between local AdS 3 × S 3 × T 4 in the IR to a linear-dilaton times S 3 × S 1 × T 4 in the UV. The dual boundary field theory interpolates between a certain vacua of LST in the UV to a CFT 2 in the IR. This allows us to systematically study holography in asymptotically non-AdS space-time. Furthermore, it has been argued that such models are intimately related to solvable models in string theory that go by the name of 'single-trace' TT deformation [32,[35][36][37][38][39]. We show that the spectrum of a single winding one long string in the interpolating coset background under investigation, exactly matches with the spectrum obtained in the case of TT deformed CFT 2 [40,41] for the particular sign of the irrelevant coupling that gives rise to a real-valued spectrum. For a generic winding w > 1 sector, the spectrum agrees with the Z w twisted sector of a symmetric orbifold theory where the block of the symmetric orbifold is obtained by TT deformation of a CFT 2 . Using world-sheet string theory techniques we compute the two and three-point functions of operators of the boundary theory in momentum space. As one moves away from the IR conformal fixed point, the operators develop a momentum dependent ambiguity in the normalization of the operators, although it is not clear to us how to fix this normalization issue. The 'dimensions' 2 of the operators at a generic point on the renormalization group flow develop a momentum dependence which resonates with the idea of momentum dependent dimensions discussed in the context of non-local CFTs [42,43].
The rest of this paper is organized as follows. In section 2 we review basic setup of superstring theory on AdS 3 × S 3 × T 4 and introduce our conventions. In section 3 we construct the main ten-dimensional coset target space-time background that we are interested in this paper. We develop thermodynamics of this background in section 4. In section 5 we perform the Kaluza-Klein reduction of the U(1) x circle and construct the nine-dimensional background, characterized by two non-trivial U(1) gauge field profiles. We construct the spectrum of physical string excitation states on top of the background fields configuration in section 6, focusing specifically on the NS-NS sector. In section 7 we explore the holographic interpretation of our results, by considering two-point correlation functions of the dual boundary theory calculated in terms of the world-sheet vertex operators in the bulk. We discuss our results and comment on future directions in section 8. Appendix A is dedicated to review of relevant background concerning (superconformal) gauged WZW models. Additional ancillary material is provided in appendices B, C.

JHEP08(2022)244 2 Review of superstrings on AdS 3 × S 3 × T 4
In this section, we will provide a brief summary of the type-II superstring theory on the AdS 3 × S 3 × T 4 background, focusing on the low-lying states in the Neveu-Schwarz-Neveu-Schwarz (NS-NS) sector. In particular, we will set up conventions related to the SL(2, R) and SU(2) Wess-Zumino-Witten (WZW) models, that we will be following for the rest of the paper. We will also review the affine Kac-Moody symmetry algebra of the target space-time, discuss its representation space, normalizability of the string states, and the corresponding vertex operators. A complementary condensed glossary of our notations can be found in appendix A. For a detailed discussion of string theory on AdS 3 we refer the reader to [4][5][6][7][8][9].
The world-sheet superstring theory on AdS 3 × S 3 × T 4 is given by the superconformal WZW model on the SL(2, R) × SU(2) × U(1) 4 group manifold. This world-sheet sigmamodel is constrained by the left-and right-moving SL(2, R) × SU(2) supersymmetric affine Kac-Moody algebra, at supersymmetric level k. In the remainder of this section, we will discuss the affine symmetry algebra and the vertex operators of the SL(2, R) ∼ AdS 3 and SU(2) ∼ S 3 subspaces separately.

Symmetries and vertex operators of AdS 3
In this subsection, we will discuss the SL(2, R) superconformal WZW model, describing the world-sheet string theory on AdS 3 . The classical supersymmetric WZW sigma-model consists of the bosonic sector, given by the SL(2, R) k+2 WZW model at the level k + 2, and three pairs of free (anti)-holomorphic two-dimensional Majorana-Weyl fermions ψ a , ψ a a = 1, 2, 3. Bosonic degrees of freedom comprise the (anti-)holomorphic Kac-Moody currents j a ,j a , a = 1, 2, 3 at the level k + 2. The fermionic sector can be bosonized, and described in terms of the SL(2, R) −2 WZW model at the level −2. Combining the bosonic and the fermionic degrees of freedom we obtain the total bosonic Kac-Moody currents forming the algebra at level k.
The exact (in the sense of world-sheet path integral) effective action for the bosonic field g ∈ SL(2, R) is given by the SL(2, R) k WZW action, where the k = 1 action S[g] is given by (A.1). The action (2.1) is invariant under the SL(2, R) L × SL(2, R) R affine symmetry algebra. The global part of this symmetry algebra is generated by the Kac-Moody currents j a andj a , a = 1, 2, 3, given by and σ a are Pauli matrices given by

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At the same time, the total Kac-Moody currents, incorporating contributions both from the bosonic and the fermionic degrees of freedom, form the (anti-)holomorphic affine algebra j a ,j a , a = 1, 2, 3 at level k, with the currents defined as 3 Using the Sugawara construction, we can write the (anti-) holomorphic components of the stress-energy tensor of the world-sheet sigma model. Focusing on the contributions from the bosonic degrees of freedom, we obtain 4 The quadratic Casimir of SL(2, R) k is given by where we defined Representations of SL(2, R) are labeled by the number j related to the quadratic Casimir as c 2 = −j(j + 1).
The spectrum of string states on AdS 3 can be constructed in terms of the unitary representations of SL(2, R). The representations relevant for studying string theory on AdS 3 can be largely classified into two groups: the principal discrete series representation 5 (lowest and highest weight) denoted by D ± j and the principal continuous series representation denoted by C α j . The lowest weight principal discrete series states (incoming states) are built on states of the form |j, j which are annihilated by j + . Thus one can write where m is the j 3 eigenvalue of |j, m . The higher weight discrete series states (outgoing states), D − j , are obtained by taking the charge conjugate of the D + j states, Normalizability of the states requires

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The principal continuous series 6 representation corresponds to states which are deltafunction normalizable, given by (2.12) where |j, α, m are eigenstates of j 3 with eigenvalue m. Without loss of generality, one can set 0 ≤ α < 1. String theory on AdS 3 also contains operators that violate the bound (2.11).
In terms of the AdS/CFT correspondence, such operators give rise to local operators of the boundary theory (see, e.g., [8] for a detailed discussion of the normalizability of the states in AdS 3 ). Next, let us review construction of the SL(2, R) primaries. The vertex operators of SL(2, R) can always be locally decomposed into a product of vertex operators of U(1) y and SL(2,R) U (1) . Such a decomposition of vertex operators is known as the parafermionic decomposition. Let us also assume that U(1) y is generated by j 3 and is parametrized by the scalar field y, satisfying the OPE y(z)y(w) − log(z − w). In terms of the field y, the current j 3 takes the form Thus, primary operator V j;m,m of string theory on AdS 3 , obtained by diagonalizing j 3 ,j 3 with eigenvalues m,m respectively, can be represented as where Ψ j;m,m is a primary vertex operator on SL(2,R) U (1) . Using the OPE of V j;m,m with the stress-energy tensor, one can read of the dimension of the vertex operator V j;m,m as This implies that the dimensions of Ψ j;m,m are given by The vertex operator V j;m,m satisfies the following OPE relations with the generators of the SL(2, R) current algebra: (2.18) 6 The continuous series states are often referred to in the literature as the long strings. Unlike the short strings, the long strings form the scattering states. They live very close to the boundary of AdS3 and have a continuous momentum.

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Switching chirality of the currents, one can write a similar set of OPE algebra of V j;m,m with the right-moving (anti-holomorphic) SL(2, R) currents. It is often useful to transform the vertex operators from the momentum space (m,m) to the position space (x,x). Let us denote the position space vertex operators by Φ j (z,z; x,x). Here, the position space coordinates (x,x) can be thought of as coordinates of the boundary of AdS 3 on which the dual CFT 2 lives. The position space representations are related to the momentum space ones via the following transformation: The operator Φ j satisfies the following reflection symmetry: Thus for real j one can restrict, without loss of generality, to j > −1/2, while for j = −1/2 + is (2.21) relates wave function with s > 0 (out-going scattering waves) to those with s < 0 (in-coming scattering waves). In the semiclassical approximation, i.e., in the limit k → ∞, the two-point function of the operator Φ h (z; x) can be determined via the world-sheet conformal invariance and the invariance under the global SL(2, R) L × SL(2, R) R symmetry of the space-time theory [8] , (2.23) and X is an arbitrary constant whose value can be adjusted by shifting the radial coordinate or rescaling the transverse field theory coordinates. In the momentum basis the two-point function takes the form . (2.24) The structure of three-point correlation function of the primary world-sheet operator Φ h is fixed by the conformal symmetry of the world-sheet and the space-time theories. For JHEP08(2022)244 the operator Φ h normalized so that its two-point function is given by (2.22), one obtains [4] can be computed from the conformal Ward identity, giving String theory on AdS 3 also contains operators obtained by a spectral flow automorphism of the symmetry algebra: whereω ∈ Z is the spectral flow parameter. Note that since we are working in the universal cover of SL(2, R), the left-and the right-moving spectral flow parameters are identical. Under spectral flow automorphism the Virasoro generators, obtained by mode expansion of the stress-energy tensor (2.6), transform as where j 3 n are the Kac-Moody modes of the current j 3 . For each spectrally flowed sectorω ∈ Z, string theory on AdS 3 contains twist operators given by The twist field tω sl acts on the spectrally un-flowed (i.e.,ω = 0) vertex operators (2.14) to generate operators in the spectrally flowed sector: It is easy to read off the world-sheet dimension of the vertex operator (2.33) as Switching the chiralities, one obtains the right-moving dimension as The OPEs of the spectrally flowed vertex operator (2.33) with the SL(2, R) currents take the form (2.37)

Symmetries and vertex operators of S 3
Superstring theory on S 3 is described by the superconformal SU(2) WZW model that is composed of the bosonic SU(2) k−2 WZW model at level k − 2 and three pairs of free (anti-)holomorphic fermions that can be described by SU(2) 2 WZW model at level 2. The bosonic and the fermionic sectors combine to give rise to the SU(2) k affine symmetry algebra at level k. For g ∈ SU(2), the exact effective SU(2) k WZW action is given by where the level-one WZW action is defined in (A.1). The action (2.38) is invariant w.r.t. SU(2) L ×SU(2) R affine symmetry algebra at level k. The global part of the SU(2) L ×SU(2) R symmetry algebra is generated by the Kac-Moody currents j a andj a , a = 1, 2, 3, given by (2.39) These currents get combined with contributions from the fermions, rendering the SU(2) k Kac-Moody algebra

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The world-sheet stress-energy tensor according to the Sugawara construction takes the form, where we omitted contributions from the fermions. The quadratic Casimir of SU(2) k is given by where All representations of SU(2) can be parametrized by j , that is related to the Casimir as c 2 = j (j + 1). The group manifold SU(2) k , being compact, possesses only discrete series representation given by where m ,m are eigenvalues of j 3 andj 3 .
Analogously to the case of SL(2, R), the primary vertex operators of SU(2) can be described in terms of the parafermionic decomposition into U(1) ysu and SU (2) U (1) . The subgroup U(1) ysu is generated by j 3 and is parametrized by a scalar field y su , satisfying the OPE y su (z)y su (w) − log(z − w). Thus one can write The primary operators, V j ;m ,m , can be written as where Ψ j ;m ,m is a primary vertex operator on SU(2) U(1) . The vertex operator (2.46) transforms in the spin (j , j ) representation of SU (2).
The SU(2) vertex operator V j ;m ,m satisfies the following OPE with the stress-energy tensor: where we defined its scaling dimension as It follows that the dimension of Ψ j ;m ,m is given by

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The OPEs of the primary vertex operator V j ;m ,m with the SU(2) currents are given by (2.50) Switching the chirality of the SU(2) currents, one can analogously write the OPE relations of V j ;m ,m with the right-moving SU(2) currents.
Similarly to the vertex operators in AdS 3 , the SU(2) vertex operators can be equivalently described in two different bases, related by The two-point functions of the SU(2) primaries are given by where the conjugate operator is given by The three-point function of V j ;m ,m is given by [44] where ∆ j = ∆ su and with the first two factors are the Wigner 3j-symbols, and

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where Π denotes (2.58) String theory on S 3 contains states obtained by spectral flow automorphism of the SU(2) algebra: whereω ,ω ∈ Z are the left-and right-moving SU(2) spectral flow parameters. Unlike spectral flow automorphism of SL(2, R), the left-and the right-moving SU(2) spectral flow parameters are different and independent of each other. Under spectral flow automorphism, the world-sheet Virasoro generators transform as where j 3 n andj 3 n are the Kac-Moody modes of the currents j 3 andj 3 . In the parafermionic notation, the spectrally flowed vertex operators are given by (2.61) Its (anti-)holomorphic scaling dimensions are given by (2.62) The OPEs of the spectrally flowed vertex operator (2.61) with the global SU(2) generators take the form (2.63) The OPE of V ω ,ω j ;m ,m with j ± andj ± remain unchanged under spectral flow.

Coset construction
The main model we are going to work with in this paper is given by the type-II superstring theory on the ten-dimensional space-time,

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given by a product of the six-dimensional coset space-time and the four-dimensional flat compact space given by the torus T 4 = U(1) 4 . The radii of T 4 are free parameters of our model. In particular, we can take the limit of infinite radii, and consider the theory on the six-dimensional coset space times the R 4 manifold.
Our primary interest will be the case of asymmetric anomaly-free time-like gauging of the U(1) sub-group in the coset sub-space of (3.1). The goal of this section is to construct the world-sheet action, obtain the gauge currents, and derive the corresponding supergravity target space-time background geometry. We will demonstrate that the target space-time geometry has a black hole horizon, while its asymptotic region is given by the two-dimensional linear dilaton background times S 3 × S 1 × T 4 . 7 We start by considering sigma-model on the group manifold that is described by the action consisting of the WZW terms for the field G ∈ G and the Polyakov terms for the abelian components. Supersymmetric version of this model is obtained by supplementing the bosonic WZW/Polyakov sector with the free Majorana-Weyl fermions. The latter are split into (anti-)holomorphic sectors, and are labeled by indices in adjoint representation of the space-time symmetry group: The world-sheet fermions are then described by the action where sums over repeated indices a = 1, 2, 3, b = 1, 2, 3, i = 1, 2, 3, 4 are implied.
As mentioned above, bosonic sector of the model is described by the sum of WZW terms for the SL(2, R) and SU(2) sectors, plus the Polyakov terms for the U(1) x and U(1) 4 subspaces. Due to the nature of the coset (3.1), it is convenient to combine the group elements into one block-diagonal element of the sub-group G of (3.2), that we denote as The WZW action for the field G on the group manifold (3.2) is then given by 7) 7 We will denote indices of vectors in ten-dimensional space-time with capital-case Latin letters, and indices of the six-dimensional coset space-time with lower-case Greek letters. E.g., the space-time coordinates are given by X M = (X µ , z 1 , z 2 , z 3 , z 4 ), where X µ are coordinates of the six-dimensional coset space-time, and z i = (z 1 , z 2 , z 3 , z 4 ) are coordinates of the torus T 4 . Throughout this paper, we will generally assume that all the coordinates are dimensionless, and therefore, e.g., the ten-dimensional metric is given by ds 2 10 = 2 s GMN dX M dX N , where s is the string length.

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where the level-one WZW action S is defined in appendix A. In terms of the field G the action (3.7) can be equivalently re-written as where we introduced the projection operator 9) and prime in the WZW action notation in l.h.s. of (3.8) indicates the presence of this projection operator. Together with the Polyakov terms for the U(1) 4 sector, total action of the supersymmetric sigma model on (3.2) is given by We view the action (3.11) as exact (in k, that also means exact in s ) effective action of the world-sheet theory. In particular, the level k of the WZW terms that we wrote down for the SL(2, R) and SU(2) sectors are exact in quantum theory. 8

Gauging and gauge currents
Next, we are going to gauge the U(1) subgroup of the group G. Let us start with the derivation in the bosonic sector, where an action of the gauged U(1) sub-group of the target space symmetry group G is defined by the following transformation laws (3.12) Here ξ parametrizes the gauged U(1) group, and x L,R are (anti-)holomorphic components of the world-sheet field x. The model is defined by the parameters a 1,2,3 , b 1,2,3 , that in particular specify how the gauged U(1) sub-group is chosen within the group (3.2). In what follows, we will impose anomaly cancellation conditions on a 1,2,3 , b 1,2,3 , rendering four independent remaining parameters. These four parameters acquire a physical meaning in terms of conserved charges of the corresponding black hole background, given by the mass and three angular momenta, as we will demonstrate below in section 4.

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We are going to be working with the following parametrization of the SL(2, R) and SU(2) group elements g = e ασ 3 e θσ 1 e βσ 3 , g = e iα σ 3 e iθ σ 1 e iβ σ 3 , (3.13) in terms of which the action (3.7) is given by (see sections A.1, A.2) while the transformations (3.12) are given by The U(1) transformation rules (3.12) can be compactly reformulated as where the left/right-moving generators of the corresponding u(1) algebra are given by the following block-diagonal matrices (3.17) Then introducing the compensator fields obeying the U(1) transformation rules we can construct U(1)-invariant field U GV . The corresponding locally U(1)-invariant WZW action can be written using the Polyakov-Wiegmann identity as

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Using (3.17) we obtain Notice that the action (3.20), while being invariant w.r.t. local U(1) gauge transformations (3.12), cannot be an action of the gauged WZW model. This is because it includes Polyakov kinetic terms (3.21) for the fields u, v, making these fields dynamical rather than auxiliary. One obvious way to eliminate these kinetic terms is given by the choice of null gauging 9 a 2 1 + a 2 2 + a 2 We are not going to pursue such a direction in this section. Instead we are going to consider the time-like gauging that ensures the anomaly-cancellation condition Tr(P T 2 L ) = Tr(P T 2 R ), thereby allowing to eliminate kinetic terms for the fields u, v by considering the action Here we introduced the U(1)-invariant field w = u − v described by the Polyakov action with the 'wrong sign'. In the time-like gauging (3.24) one easily derives where we denoted the auxiliary gauge field potentials as Furthermore, introducing the U(1) L,R gauge currents we can re-write the action (3.25) as Eliminating the non-dynamic fields A,Ã via e.o.m., we obtain the action

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and a non-trivial dilaton field Using the parametrization (3.13) we obtain The dilaton background (3.32) can then be determined as For the currents (3.28) we obtain where we defined (recall that σ a are Pauli matrices) (3.37) In the parametrization (3.13) we obtain (3.38) We will also use the notations The current components j 3 and j 3 , and their anti-holomorphic counterparts, belong to the sl(2, R) and su(2) Kac-Moody algebras at levels k + 2 and k − 2, satisfying the OPEs 10

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where abc is the Levi-Civita symbol, 123 = 1, and similarly for the anti-holomorphic components. The currents j x , j i , and j w are abelian Kac-Moody currents at levels 1, 1, and −1 respectively, 11 (3.42) and similarly for the anti-holomorphic sector.
In the supersymmetric theory (3.11) one additionally has contributions to the gauge U(1) L × U(1) R currents (3.36) coming from the fermionic sector. The total gauge currents can be compactly reformulated in terms of the total bosonic currents of supersymmetric Kac-Moody algebra, 12 that in particular implies Here the sl(2, R) and su (2) fermions satisfy the OPEs For the total bosonic currents, we have the following OPEs of the supersymmetric Kac-Moody algebras sl(2, R) and su(2) (3.46) both at level k. The supersymmetric theory counterpart of (3.36) is then given by Taking into account anomaly-cancellation conditions (3.24), we then obtain Supersymmetric partners of the gauge currents (3.47) are given by the fermions These follow from the free field OPE (3.41) More comments on the sign of the ∂w∂w two-point function will be given in section 6.1. 12 See appendix A.3 for a review.

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Using (3.45) in combination with the OPEs we notice that the gauge fermions (3.49) that satisfy the OPEs

Supergravity background
Combining (3.14), (3.34), (3.47), (3.38) in (3.31) we can write down the full world-sheet gauged WZW action for the considered parametrization (3.13). This action is invariant w.r.t. gauge transformation (3.15). We can fix the associated redundancy in the degrees of freedom by the gauge choice where ν is an arbitrary parameter. For the sake of calculating supergravity background geometry, we set all the fermionic d.o.f. to zero. The resulting action is then given by where we defined the world-sheet field defining coordinates in ten-dimensional target space with the non-vanishing components of the metric given by the six-dimensional coset subspace components

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as well as the components of the metric on the torus T 4 , given by Non-trivial components of the B-field are given by that can be derived from the corresponding bulk gravity action in the string frame and the dimensionful volume of the four-dimensional compactified torus (in string frame) is given by (here R i are dimensionless radii of the circles of The corresponding metric on target space-time is given by

64)
13 Recall that the B-field is defined up to gauge transformations that leaves the tensor (3.62) invariant.

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where we restored the string length s , taking into account that in our conventions the space-time coordinates X M are dimensionless. The metric in Einstein frame,G M N , is related to the metric in string frame, G M N , via In Einstein frame normalization of the gravity action is given by 66) and G N is the Newton's constant. 14 Notice that due to (3.65) the volume of the four-torus in the Einstein frame is given byṼ To study the background (3.55), (3.56), (3.35) it is convenient to choose a different coordinate frame. Let us begin by rescaling the U(1) x coordinate as follows, Then every component of the metric and the B-field will be proportional to k, and therefore dependence of these tensors on the WZW level can be compactly expressed through an overall common factor. Next, we rescale the time coordinate y as follows, 15 We also introduce a new radial coordinate ρ(θ) according to the rule (recall that ∆(θ) was defined in (3.34)) where we defined For the choice of parameters the coordinate transformation (3.69) therefore implies the range of validity of the ρ coordinate given by In the new coordinates we then obtain the dilaton 14 See, e.g., [45]. 15 While this choice of the time coordinate can appear ad hoc at this point, the rationale behind it will be explained below, when we will show that in such a frame one obtains adjusted asymptotic valueĜtt/k = −1.

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and the metric Here we have defined and used G ab , a, b = α , β , x to denote projection of the G µν tensor on the three-dimensional sub-space spanned by the vector (3.75). We have also re-arranged the terms in (3.74) in such a way that the off-diagonal metric elements G tα , G tβ , G tx are expressed in terms of the vector where G ab is the inverse metric on the three-dimensional sub-space spanned by (3.75). We then obtain the adjusted metric component we can expressĜ We therefore observe that ρ 2 = ρ 2 + is the outer horizon of the metric. Notice that ρ 2 16 Therefore the apparent singularities ofĜ tt at ρ 2 = 2 ± are hidden behind the outer horizon ρ 2 = ρ 2 + . Let us define where we separated explicitly asymptotic values at ρ → ∞, leaving us with 16 This is easiest to see by taking a1 = cos ψ, b1 = cos ψ .

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By performing the change of coordinates we can remove the off-diagonal (t,α ), (t,β ), (t,x ) metric elements in the asymptotic region. In the new coordinates we obtain The rest of the components of the metric are given by (3.85) Notice that dependence on the arbitrary gauge parameter ν completely disappeared. In the asymptotic region ρ → ∞ we obtain the linear dilaton background times S 3 × S 1 where the SU(2) ∼ S 3 metric is given by 17 (3.88) 17 By performing the change of coordinatesα = (φ2 − φ1)/2,β = (φ2 + φ1)/2, we can bring this metric to the Hopf form ds 2

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By picking a new radial coordinate χ we can eliminate the G χθ off-diagonal term in the metric tensor. For instance, let us choose The new radial coordinate in the domain of parameters (3.71) is then defined on the interval and χ + represents the outer black hole horizon. The interval in the new coordinates is given by In the new coordinates we can also express where we denoted The metric (3.91) defines a rotating black hole with the event horizon at χ ≡ χ + , θ ≡ const, dX a = −R a dt, a =α ,β ,x . The latter indicates that the angular velocity at the horizon is given by 19 Using (3.93) we then obtain (3.97) 18 In Hopf coordinates (3.87) we have G φ 1 φ 2 = 0. 19 Here the factor of √ k originates from the fact that our time coordinate t is normalized so that the asymptotic behavior of the metric is ds 2 = −k 2 s dt 2 + . . .

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These can be further split into the left-and right-moving components The full metric can also be written as (recall that 0 is given by (3.95)) 20 The dilaton is given by The Ricci scalar is given by (3.102) Finally, for the independent components of the anti-symmetric B-field we obtain Notice that the unphysical ν-dependent terms can be readily eliminated using the gauge , for the following choice of transformation parameters (3.104) 20 Setting a1 = cos ψ, a3 = sin ψ, b1 = 1, a2 = b2 = b3 = 0, changing the radial coordinate to χ = 2 e 2r cos ψ − 1, and performing Kaluza-Klein reduction of the x coordinate we recover the metric of twodimensional charged black hole in string frame, The dilaton is then given by Φ = Φ0 − 1 2 log(2 cos ψ)−r, in agreement with [30].

Conserved charges
The black brane background (3.100), (3.101), (3.103) possess symmetries associated with translations in time t, angular coordinatesα ,β of the S 3 and compact coordinatex of the S 1 . In this subsection, we are going to derive the corresponding expressions for the conserved energy and angular momenta. Angular momenta components are defined via the 1-forms by the Komar formula (see, e.g., [46,47]) in string frame as the linear combinations In (3.106) the star denotes the usual Hodge dual of a p-form, and τ χα θ β x = 1. In (3.106) the integral is performed over the asymptotic boundary S 3 × S 1 , which selects the corresponding component (α θ β x ) of the 4-form (dk a ). We then obtain (3.109) For the energy/mass we can analogously use the Komar formula (3.111) 21 In general, adding constants to the components of the B-field cannot affect supergravity equations of motion, that depend on the B-field via the tensor H = dB. 22 Here the factor of √ k is due to our choice of normalization of the time coordinate, which is dual to the energy E, according to which ds 2 = −k 2 s dt 2 + . . . in the asymptotic region. 23 Recall that relation between κ and κ0 is given by (3.66).

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In the next section, we will demonstrate that the derived expressions for energy and angular momenta are consistent with the first law of thermodynamics.

Thermodynamics
In this section, we are going to study thermodynamics of the background (3.100), (3.101), (3.103). This background describes a black brane, rotating with non-vanishing angular velocities along two of the S 3 as well as the S 1 directions. We are going to determine the temperature and derive the associated Bekenstein-Hawking entropy. Using expressions for the energy, angular velocities, and angular momenta calculated in the previous section, we will derive the free energy, and demonstrate explicitly the first law of thermodynamics. Additionally, by explicit calculation of the regularized on-shell action, we will demonstrate that the free energy vanishes, in agreement with the general expectation for an asymptotically linear dilaton space-time.
The near-horizon limit of the metric (3.91) can be taken by expanding around r = 0. This gives The null interval at the horizon is then defined by while regularity at the horizon requires that Euclidean time has the period expressed in terms of the black hole temperature as τ ∼ τ + 1/T . Recall that our Lorentzian time coordinate t is normalized so that the asymptotic metric at the boundary is determined by ds 2 = −k 2 s dt 2 + . . . . Then the temperature is given by The temperature (4.4) takes values in the range is the Hagedorn temperature of two-dimensional black hole. The zero-temperature limit, T = 0, is achieved for the choice of parameters a 1 b 1 = 0, in which case we obtain χ + = χ − . In the limit a 1 → 1, b 1 → 1 we get T → T H , which corresponds to sending the outer horizon to asymptotic linear dilaton region of space-time.
At the outer horizon we obtain for the Einstein frame metric where i, j =α , θ ,β ,x , and G ij is projection of the metric on the associated sub-space, and we substituted expression (3.101) for the dilaton. The horizon area, obtained by integrating over θ ∈ 0, π 2 , α ∈ [0, π], β ∈ [0, 2π] is therefore given by

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where we also denoted dimensionful length of the U(1) x circle (in string frame) as following from the periodicity condition for the dimensionless coordinate x ∼ x + 2πR x . The Bekenstein-Hawking entropy is therefore given by 24 Now let us proceed to calculation of the Euclidean on-shell action where S 0 is obtained by plugging in the background solution into the bulk action (3.60), S GHY is the Gibbons-Hawking-York (GHY) boundary term, and S c.t. is the counter-term action. Let us start with the former. Using (3.58) we can re-write (3.60) on shell as 25 (4.12) Using the Stokes's theorem, we obtain where is the metric on the boundary χ = Λ, 26 and n µ , given by (B.7), is the outward-pointing unit vector normal to the boundary. We can then re-write (4.13) as where, as before, det G is determinant of the six-dimensional metric. Using (3.100), (3.101), we obtain for any χ, One can parametrize the asymmetric gauging of the U(1) sub-group as (see [32] and references therein) i.e., decoupling the SU(2) sub-group. Taking the same limit in (4.10), we can compare it with the entropy of rotating black string following from the expressions for the metric Gxx and dilaton Φ − Φ0 given by eq. (3.36) in [32]. It suffices to point out that the horizon area is given by that after a simple manipulation one can see to coincide with (4.6) for the parameters (4.8). 25 In the second line we have applied . 26 The corresponding contribution from the lower limit of integration χ = χ+ vanishes.
The GHY term at the boundary χ = Λ in string frame is given by (B.10), In (4.20) we used the induced metric (4.14) on the boundary χ = Λ. At the same time, K = ∇ µ n µ is trace of the extrinsic curvature tensor, defined by the normalized outwardpointing unit vector (B.7) normal to the surface χ = Λ. We then evaluate that finally gives consistently with the Hagedorn growth of density of states, and in agreement with analogous result for the two-dimensional charged black hole [48][49][50].
On the other hand, the free energy is given by where we introduced short-hand notation where components of the angular velocity have been derived in (3.98), and angular momenta are given by (3.109). Using these expressions and taking into account the anomalycancellation constraints, we derive

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Combining expressions for the entropy (4.10), angular velocities and momenta (4.27), and mass (3.111) in the expression for the free energy (4.25) we can explicitly reproduce that the free energy vanishes, in agreement with (4.24). We can also verify that the first law of thermodynamics is satisfied, This identity can be demonstrated for arbitrary da 1,2 , db 1,2 by substituting Ωα R = Ωβ L = 0, using dE = 0, and taking into account that anomaly cancellation conditions imply Lastly, we notice that the four independent parameters defining the gauging of the U(1) sub-group that remain after the anomaly-cancellation conditions have been imposed on a 1,2,3 , b 1,2,3 , can be expressed in terms of the temperature and angular momenta.

Two-charge background
In section 3.2, we derived the metric, dilaton, and B-field ten-dimensional background configuration (3.100), (3.101), (3.103) for string theory on the coset space (3.1). This background possesses an event horizon, that rotates with the angular velocities given by (3.97) in the directions ofα ,β ,x . It asymptotes to a static linear dilaton two-dimensional space-time times S 3 × S 1 × T 4 . In this section, we are going to construct the nine-dimensional background obtained from (3.100), (3.101), (3.103) by compactifying the U(1) x coordinatex . To be precise, due to the non-vanishing angular velocity Ωx at the horizon, we first need to perform a change of coordinatesx →x , defined by 27x such that the new coordinatex is not rotating at the horizon. By performing the KK reduction ofx , from the ten-dimensional background G M N , B M N , Φ, we can derive the nine-dimensional background where we chose indices m, n to label coordinates of the nine-dimensional non-compact background. This background features two U(1) gauge fields A m , B m , that are sourced by the corresponding charges Q A , Q B at the chemical potentials µ A , µ B that we will calculate below. 27 An extra factor of 1/ √ k is due to our conventions of separating an explicit common factor of k in expression for the interval ds 2 .

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We can write the five-dimensional coset sub-space metric of the nine-dimensional background in the form analogous to (3.91), where indices r, s take valuesα ,β , the two-dimensional metric G rs is given by , , and we have definedRα In the asymptotic region χ → ∞ we have a static background,R r = O(1/χ), while the metric G rs approaches The nine-dimensional background then asymptotes to a static linear dilaton twodimensional space-time times S 3 × T 4 . We can calculate angular velocities at the horizon according to Ω r = − √ kR r | χ=χ + , r =α ,β , analogously to (3.96), yielding in agreement with (3.97). We can determine the corresponding angular momenta J (G,B) r , r =α ,β using (3.106) applied to the nine-dimensional background, rendering in agreement with (3.109).
One can see that at the horizon the time-like components of the gauge fields in corotating reference frame vanish,Ã t | χ=χ + = 0,B t | χ=χ + = 0, up to a gauge transformation. Recall that since the horizon is rotating in the directions ofα ,β , we actually need to calculate gauge fields in the co-rotating coordinateŝ where we have also performed a constant shift The remaining non-trivial components of the gauge fields are given bỹ From asymptotic behavior of the gauge potentials we can read off the chemical potential that we can subsequently split into the left-moving and right-moving components as Expressions for the chemical potential (5.15) agree with their angular velocities counterparts Ωx L,R before compactification, given by (3.98), The corresponding charges Q L,R can be found analogously to [51,52]. We first calculate where the field strength tensors are defined as F rs Notice that these expressions match the angular momenta J G,B x of the ten-dimensional background (3.109). The corresponding left-and right-moving charges can then be found according to

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The free energy of the nine-dimensional background is given by expression analogous to its ten-dimensional counterpart (4.25), except that now we have rotations only inα ,β directions, as well as two U(1) charges, Here we can calculate in agreement with (4.27). The values for temperature T , entropy S, energy E, and grand potential F of the nine-dimensional background are the same as for the ten-dimensional background, and are given by (4.4), (4.10), (4.24), (3.111). Finally, one can explicitly verify that the first law of thermodynamics is satisfied,

Spectrum
In this section, we will discuss construction of spectrum of type-II superstring theory on the coset space (3.1). As we will review below, physical states of superstring theory satisfy the super-Virasoro constraints, ensuring superconformal invariance of quantum theory on the world-sheet. Additionally, physical states of the coset sigma-model satisfy gauge-invariant conditions, that can be formulated via the BRST formalism [53]. 28 We will derive the corresponding constraints in section 6.1.
It is important to emphasize that our conventions in this section are to use the coset BRST formalism to impose the U(1) L × U(1) R gauge symmetry in the spectrum of quantum theory. At the same time, we will use the covariant quantization approach (without employing the superconformal BRST formalism) to ensure superconformal invariance of the spectrum in terms of the (super-)Virasoro constraints.
We are mostly interested in the bosonic excitations of the NS-NS sector, in particular, the lowest tachyon state, and the massless states composing the gravity multiplet of the type-II supergravity. Therefore the distinction between type-IIA and type-IIB superstrings, originating from the choice of space-time chirality of the fermionic states in the R-NS and NS-R sectors, will not play a role in most of the subsequent discussion. We will also not consider bosonic excitations coming from the R-R sector, leaving these topics for future work.

Coset BRST constraints
In this subsection, we are going to derive the coset BRST constraints, that have to be satisfied by the physical states on the coset (3.1). These constraints impose the U(1) L × U(1) R symmetry on the states of the physical spectrum of the coset theory. (We stress once again, that in this paper the superconformal BRST formalism is not used to impose superconformal symmetry on the physical states.)

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To lighten up the discussion, we will begin by performing derivation of the BRST constraints for the six-dimensional coset subspace of the target space-time (3.1), working in the purely bosonic sector. We will subsequently formulate the supersymmetric version of the obtained constraints. Partition function of the gauged WZW model (3.29 Ghost fields satisfy the following non-trivial OPEs The ghosts b and c are conformal primaries of dimensions 1 and 0 respectively [16,54], 29 with the following expansions in integer modes, Conformal dimensions of the ghosts can also be reproduced from their the stress-energy tensor, T bc (z) = b(z) ∂c(z) , (6.8) where the product in the r.h.s. is normally-ordered. Finally, let us change integration variables u → u + w, G → U −1 GV −1 and divide the partition function (6.2) by the constant factor of [Dv], rendering 29 As a side note, in the BRST quantization approach to superstring theory one employs the superconformal ghosts comprised of the fermions b, c, and bosons β, γ. Dimensions of these pairs of ghosts are (2, 1) and (3/2, −1/2), respectively. In particular, this is reflected in contributions of b, c to the stress-energy tensor, Tbc = −2 b ∂ c+c ∂ b, which is manifestly different from its coset BRST ghosts counterpart (6.8). We do not perform superconformal BRST quantization of superstring theory, and therefore the superconformal ghosts b, c, β, γ do not appear in our derivation of the spectrum.

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where the full action of the model is given by Notice that from the action (6.10) it follows that the two-point correlation function of the field w has the 'wrong' sign This does not pose a problem, since the physical state (coset BRST) conditions will eliminate the corresponding negative norm states from the spectrum. These conditions are derived as follows. Consider the holomorphic BRST transformation defined by the fermionic parameter η, with the non-trivial variations given by [55] δG = iη c T L G , δw = iη c , δb = iη (J + 2∂w) , (6.12) where the gauge current J is given by (3.28). Using (A.12) we can derive the corresponding variation of the action (6.10) as Analogous calculation can be done for the anti-holomorphic BRST transformation, δG = −iηc GT R , δw = iηc , δb = iη (−J + 2∂w) , (6.14) giving variation of the action For a global parameter η the transformations (6.12), (6.14) are therefore a symmetry of the action (6.10). Choosing a local parameter η we obtain the coset BRST currents J BRST = c J (0) ,J BRST =cJ (0) , where we denoted the corresponding null currents as 30 Let us now switch to discussion of the full supersymmetric model. To begin with, recall that the U(1) L × U(1) R currents J,J of the purely bosonic sector, given by (3.28), are to be replaced with their supersymmetric theory counterparts J,J, given by (3.47). These total bosonic gauge currents incorporate contributions from the fermions in adjoint representation of the sl(2, R) and su (2) algebras. The corresponding null currents in the supersymmetric coset theory are then given by (6.17)

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Indeed, combining (3.48), (6.11) we recover the BRST null conditions The associated BRST charges of the currents J BRST = c J (0) ,J BRST =cJ (0) are therefore nilpotent, Q 2 BRST = 0,Q 2 BRST = 0. Physical states in the spectrum of the coset model (6.10) are to be closed w.r.t. these charges, Q BRST |phys = 0 ,Q BRST |phys = 0 , (6.20) and are defined up to the BRST-exact states. Expanding the null currents (6.17), −1 |0 , where |0 is the physical vacuum. Using (6.18) we also obtain the commutation relations The (anti-)holomorphic superpartners of the gauge currents J,J are given by the fermions (3.49). At the same time, the superpartners of the bosonic currents j w ,j w are given by the fermions ψ w ,ψ w , such that 31 We can use these to write down the fermionic null currents superpartners [56] 32 that satisfy the null conditions Physical states in the spectrum of NS states of the supersymmetric coset theory should satisfy the constraints , . . . , (6.27) 31 Notice that the sign of these correlation functions is opposite to the sign of the two-point functions of physical fermions (3.50). This is analogous to having opposite signs of physical and auxiliary bosonic currents (3.42). Unphysical w polarization is eliminated from the spectrum by the BRST constraints. 32 The fermionic sector possesses its own ghosts, which are bosonic counterparts of b and c. We skip detailed discussion of this formalism.

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where we defined the NS sector mode expansion amplitudes as where r is half-integer in the NS sector. Using (6.26) we obtain

Virasoro constraints
In section 6.1, we used the coset BRST formalism to derive the U(1) L × U(1) R null physical state constraints J −r |0 where |0 is the physical vacuum. Besides these constraints, the physical states of superstring theory have to satisfy the (super-)Virasoro constraints. These constraints reflect superconformal invariance of the quantum world-sheet theory. The superconformal transformations are generated by the stress energy tensor T (z), and the super-current G(z), and their anti-holomorphic counterpartsT (z),G(z). These currents form (anti-)holomorphic affine superconformal algebra, that we briefly review in appendix A.3.
Components of the stress-energy tensor of the coset world-sheet theory (3.1) are given by where T T 4 ,T T 4 are the components of the world-sheet stress-energy tensor with target space T 4 parametrized by coordinates z i , i = 1, 2, 3, 4, iji . (6.31) We have also denoted contributions from the fermionic sector as and similarly for the anti-holomorphic sector, while contributions from coset BRST ghosts, T ghosts comes from the ghosts sector, that includes T bc given by (6.8), and its supersymmetric counterpart. The (anti-)holomorphic supercurrents G(z),G(z) are fermionic operators. We are mostly interested in the NS sector, and therefore the fermions are expanded in half-integer modes. Specifically, expanding the stress-energy tensor and the supercurrent, we obtain Additionally, the physical states are defined up to exact states L −n |0 = 0, G −r |0 = 0, and similarly for the anti-holomorphic sector.

Ground state vertex operator
We now proceed to construction of the primary vertex operator of the (3.1) coset model. This operator describes the ground state of the model, that we write as where W (0) (z,z) is the ground-state vertex operator of the six-dimensional coset sub-space of (3.1), while 34 is due to the four-dimensional flat sub-space T 4 = U(1) 4 . The corresponding ground state V (0)Ṽ (0) |0 should satisfy physical conditions, that include the coset BRST constraints and the superconformal super-Virasoro constraints. In this subsection we will focus on the former.
We start by expressing the coset ground state vertex operator W (0) (z,z) as a product of the SL(2, R), SU(2), U(1) x , and U(1) w 35 primary operators, whereω ∈ Z is the SL(2, R) spectral flow parameter andω ,ω ∈ Z are the SU(2) leftand right-moving spectral flow parameters. The left-and right-moving momenta on the compact U(1) x circle with radius R x are given by where n and w are the momentum and winding numbers respectively. Recall that the SL(2, R) and SU(2) primary operators satisfy (2.37), (2.63) 39) 33 In terms of the vertex operator V corresponding to the state |phys , one calculates, e.g., Ln · V(0) 1 2πi z n T (z) V(0) and sets it to zero. 34 Recall that in our conventions all the coordinates are dimensionless. At the same time, we will define all the momenta to be dimensionful. 35 The U(1)w is non-compact, and therefore κL = κR = κ.

Asymptotic behavior of vertex operators
In this subsection, we are going to explore asymptotic behavior of vertex operators describing string excitation modes. We begin with the ground state vertex operator (6.37). Specifically, we are interested in the plane-wave behavior in the asymptotic region, that allows one to read off energy of the string excitation mode. In fact, the same energy characterizes any string excitation obtained by acting with the creation operators on the given ground state vertex operator. In the SL(2, R) parametrization given by (3.13), the asymptotic region is found by taking the limit θ → ∞. The SL(2, R) ground state vertex operator in this limit behaves as [28] Vω j;m,m (θ → ∞) e iy[(m+ kω 2 )ν+(m+ kω 2 )] e 2θj + (−1) −(2j+1) R(j; m,m)e −2θ(j+1) , (6.44) where (this coincides with the coefficient in (2.24)) . (6.45) In the parafermionic decomposition one represents locally SU(2) ∼ SU(2) U(1) × U(1) y , that allows one to re-write the SU(2) ground state vertex operator as where Ψ j ;m,m is the SU(2)/U(1) vertex operator.

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Combining expressions (6.44), (6.46) for the SL(2, R) and SU(2) ground state vertex operators in the expression (6.37) for the ground vertex operator of the coset sub-space of (3.1) and focusing on the plane-wave phase in the asymptotic region, we obtain Here we can substitute expressions for m,m, obtained by solving the BRST constraints (6.42), which after some re-arrangement gives where we defined the time coordinate t (6.49) as well as the moving frame coordinateŝ Notice that the time coordinate change (6.49) and the moving frame coordinates change (6.50) agree with the corresponding expressions (3.68), (3.82) obtained by analyzing the asymptotic region in derivation of the coset background geometry. 36 Defining time as t → t √ k − s w, we conclude from the asymptotic plane-wave expression (6.48) for the vertex operator W (0) that the frequency of the excited string state defined by this vertex operator is given by κ. Notice that this is true regardless of whether we consider a spectrally flowed sector.

Mass-shell condition and spectrum
In subsection 6.3, we constructed ground state vertex operator (6.35) of the coset model (3.1), and derived coset BRST physical state constraints (6.42) that such an operator needs to satisfy. In subsection 6.2 we wrote down the (super-) Virasoro constraints that need to be satisfied by the (NS sector) states of superstring theory on the coset (3.1). Only a finite number of non-trivial constraints, depending on the string excitation number, need to be imposed for each state, with the rest of infinitely many constraints satisfied automatically. However, every state needs to satisfy the lowest Virasoro L 0 ,L 0 constraints, known as the mass-shell condition. In this subsection we will derive the mass-shell condition for string states on the coset (3.1).
Let us denote the (anti-)holomorphic superstring vertex operator at the level (N ) N as (Ṽ (N ) ) V (N ) . It is obtained by acting on the ground state vertex operator V (0)Ṽ (0) (that 36 The relative factor of √ k in the transformation laws for x = xL + xR is due to the relation between x and x given by (3.67).

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corresponds to N =N = 0) with the product (αN ) α N of creation operators such as (ψ M −r , (6.51) such that 37 Using (6.52) together with the OPE of the ground state vertex operator (6.35) with the stress-energy tensor, we derive (6.53) where N,N = 1, 2, . . . are the string excitation numbers, are scaling dimensions of spectrally-flowed ground state vertex operators of the SL(2, R) and SU(2) WZW models (see (2.35), (2.36), (2.62)), and are scaling dimensions of the ground state vertex operator (6.36) on T 4 . The mass-shell condition, given by the L 0 ,L 0 Virasoro constraints of (6.34), then reads In section 6.4 we established that the energy of string state, defined by the planewave behavior of the vertex operator in the asymptotic region, 38 is given by κ. The L 0 ,L 0 Virasoro constraints (6.56) then have the physical interpretation of the mass-shell conditions, with the terms in the r.h.s. of these expressions given by the sum of the momenta (on SL(2, R) and SU(2) the role of momenta are played by the representation numbers j, 37 See (A.65) for commutation relations of the Virasoro amplitudes with the super-Kac-Moody currents. 38 The plane-wave calculation is true for any excitation numbers N =N , and is determined only by asymptotic behavior of the ground state vertex operator W (0) .

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m, j , m ) and oscillator numbers N − 1/2,N − 1/2. Substituting (6.54) in (6.56) and imposing the BRST constraints (6.42) to eliminate m,m, one obtains 39 For the ground state vertex operator (6.35) in the NS-NS sector we have N =N = 0. 40 The state is then tachyonic, κ 2 < 0, at zero momenta. Such a state is removed from the physical spectrum by the GSO projection. The first excited states of the NS sector, with (N = 1 2 ) N = 1 2 defined by the (anti-)holomorphic vertex operators (Ṽ ( 1 2 ) ) V ( 1 2 ) are massless, and represent the lowest-lying state in the physical spectrum of string excitation. Irreducible representations of tensor product of these states form the NS-NS sector of the supergravity multiplet, and will be studied below in section 6.6.

Massless states
Massless states of the type-II superstring theory form the type-II supergravity multiplet. Half of its bosonic degrees of freedom are contained in the NS-NS sector, and form a gravity multiplet, consisting of the graviton, dilaton, and B-field. In section 3, we have derived background values of these fields, (3.100), (3.101), (3.103), for superstring theory on the coset (3.1), exactly in k.
In this subsection, we are interested in excitations of the massless NS-NS fields on top of the background (3.100), (3.101), (3.103). Performing the KK reduction on the U(1) x circle, one can also describe excitations on top of the nine-dimensional background (5.2). These excitations can be obtained by first separately constructing (anti-)holomorphic massless vertex operators in the NS sectors of the (right-) left-moving string states, and then taking the direct product of these vertex operarors. Finally, the resulting closed string states can be grouped into irreducible representations of the target space-time symmetry group, rendering the graviton, dilaton, and B-field excitations.
The massless holomorphic vertex operator V ( 1 2 ) in the NS sector on the coset (3.1) is obtained by acting with the creation operators ψ M −1/2 on the tachyon ground state vertex operator (6.35). For the most general polarizations ξ M in the target space-time we obtain JHEP08(2022)244 the state where we introduced linear combinations of the sl(2, R) and su (2) fermions In (6.58) we have chosen to replace the basis target space direction ψ 3 with the null spinor ψ (0) given by (6.25). Physical state conditions, that include the coset BRST constraints and super-Virasoro constraints, and remove the pure-gauge exact states, restrict the allowed polarizations ξ M , rendering eight independent polarizations. Due the coset BRST constraints, the state (6.58) needs to be closed w.r.t. the null spinor ψ (0) , given by (6.25). In terms of the amplitudes (6.28) this condition imposes In terms of the basis fermionic operators, this can be written explicitly as The BRST constraint (6.61) can be solved as The most general massless holomorphic BRST closed state is then given by Here the polarization ξ is un-physical, and represents a pure-gauge exact state on the coset. After imposing the coset BRST physical state conditions, the vertex operator (6.63) then possesses ten independent polarizations. Next, let us discuss the super-Virasoro constraints (6.34). For the massless state, V ( 1 2 ) , the only non-trivial constraints are given by and its counterparts in the anti-holomorphic sector. The L 0 constraint is simply (6.56) with N =N = 1/2, giving the mass-shell condition for the massless state. The
In preparation for discussion of holography in section 7, we are now going to consider (anti-)holomorphic massless vertex operators polarized along the direction of propagation JHEP08(2022)244 of excitations, as well as along the U(1) x circle. Without loss of generality, we can define the former to be the coordinate z 1 on the torus T 4 , p 1 L = p 1 R = p, setting the rest of the momenta on the torus to zero, p i L,R = 0, i = 2, 3, 4. We are also going to perform the KK reduction on the U(1) x circle, setting the corresponding momenta to zero, p L = p R = 0. (Anti-)holomorphic vertex operators polarized along the U(1) x circle will then be found as constituents of the gauge field vertex operators of type-II superstring theory in the nine-dimensional space-time.
In other words, we will be looking for the physical massless vertex operators of the form Ψ 1 we will fill in the missing terms accordingly with the expression for the physical vertex operator (6.63), (6.70). Specifically, in the former case we set ξ 1 = 1, ξ x = 0, while in the latter case we set ξ x = 1, ξ 1 = 0. We also set ξ 2,3,4 = 0, ξ ± = 0, ξ 3 = 0,ξ = 0 everywhere. For convenience, we will denote In general, physical massless states can then be defined as With such a convention, physical massless states are constructed using creation opera- , that automatically take into account the coset BRST and super-Virasoro constraints. Using (6.63), (6.70), we then obtain and similarly for the anti-holomorphic components. Using (6.66), (A.67), we obtain the two-point correlation functions where we have denoted , (6.77) and used a short-hand notation · · · 0 for averages in the state V (0)Ṽ (0) |0 . Analogous correlation functions can be written down in the anti-holomorphic sector. The tachyon twopoint function V (0) V (0) is given by a product of the two-point functions of its constituent JHEP08(2022)244 vertex operators (6.35), (6.37). For the full state, including both the anti-holomorphic and the holomorphic sectors, we obtain where one should substitute (2.24), (2.53) for the two-point functions of the SL(2, R) and SU(2) ground state vertex operators. Notice that zero of (2.24) at m = −j cancels out the simple pole of (6.77) at m = −j. At the same time, F j;m has a simple pole at m = j, which is inherited by all the correlation functions (6.76). Similarly, the anti-holomorphic sector has a simple pole atm = −j, while exhibiting regular behavior atm = j.

Spectrum of single-trace TT -deformed symmetric product CFT
We will end this section with a discussion of the spectrum of the long strings in the coset model (3.1), and its relation to the spectrum obtained by a deformation of CFT 2 by the TT operator. As it turns out, the spectrum of a single long string in the sector with the unit winding number, w = 1, corresponds to the spectrum of the double-trace TT deformed CFT [32]. For a detailed review of the single-trace TT deformation of string theory in AdS 3 , and its relation to the long strings, we refer the reader to [37]. Let us begin by reviewing certain aspects of the theory on a single long string in AdS 3 . Via the AdS/CFT correspondence, string theory in AdS 3 × N 7 is dual to a CFT 2 living on the boundary of AdS 3 . 41 Here N 7 stands for a seven-dimensional compact space, e.g., In the presence of pure NS-NS flux in the bulk, the space-time theory has an SL(2, C)-invariant vacuum. This is either an NS vacuum, that corresponds to global AdS 3 in the bulk, or an R vacuum, that corresponds to a massless BTZ (denoted by BT Z M =J=0 ). The NS sector contains states that belong to the discrete as well as the continuous series representation of SL(2, R). The continuous series representation starts above a gap of order k/2. The R sector, on the other hand, contains a continuum of long strings above a gap of order 1/k. In the rest of the discussion of the spectrum of long strings in this section, we will consider only the long string states in the R sector.
On the AdS 3 × N 7 with pure NS-NS flux, the theory on a single long string is given by the product of the N = 2 Liouville theory R φ and the compact SCFT on N 7 [57] M (L) where c M = 6k is the central charge of M (L) 6k and the theory on R φ has a dilaton linear in the radial direction of AdS 3 denoted by φ with the slope The effective coupling on the long strings is given by

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implying that for k > 1, the long strings become strongly coupled as they move towards the boundary. But there is a wide range of positions along the radial direction where they are weakly coupled. The effective theory of N such long strings is described by the symmetric product [37,58] (M Let us now proceed to discussion of long strings on the coset (3.1). Recall that the value of κ is equal to the energy of a string excitation mode propagating in the coset background. The long strings in that background belong to the continuous representation of SL(2, R), for which we define, due to (2.12), For a non-vanishing winding w around the U(1) x -circle, the energy E of such a long string in the spectrally un-flowed sector (i.e.,ω =ω =ω = 0), above the BPS configuration is given by where due to (6.57) we have (6.85) Due to the dispersion relation of the long strings in the winding w sector in BT Z M =J=0 × S 3 × T 4 that is given by [59] 86) expression (6.85) can be written as Expression (6.86) also gives the spectrum of the Z w twisted sector of the symmetric product CFT M N /S N where the block CFT M is the CFT with central charge [58] c M = 6k (6.88) living on a single long string with winding one. This is not surprising because the effective theory on the long strings are well described by a symmetric product (6.82) at weak coupling [58]. Spectrum of operators of dimension h w in the Z w twisted sector of the symmetric product CFT in the Ramond sector above the Ramond vacuum is given by [58,60] For every operator of dimension h 1 in the untwisted sector there exists an operator in the Z w twisted sector whose dimension is related to h 1 as [60] Next we solve (6.84) for κ in terms of E, and we also substitute (6.90) into (6.89). Plugging the result into (6.87) one obtains [32,[35][36][37]] In the untwisted sector corresponding to w = 1, (6.91) takes the form [40,41] which is the spectrum of a CFT 2 on a cylinder of radius R x deformed by an irrelevant operator δL = −t TT with t = 2 s > 0. For w > 1, (6.91) is the spectrum of the Z w twisted sector of single trace TT deformation of a symmetric product CFT [61].

Holography
In this section, we are going to derive various two-point correlation functions of the field theory holographically dual to the bulk string theory on the coset (3.1). We are going to explore different regimes of the holographic correspondence, and calculate various correlation functions in the NS-NS sector of string theory in the bulk.
In subsection 7.1, we will consider the bulk configuration, derived in section 5, obtained by compactification of the U(1) x circle of the coset (3.1). This background is characterized by a rotating black brane geometry with an event horizon, finite temperature, and two independent non-trivial U(1) gauge field profiles. We will study massless excitations modes described by the metric, B-field, dilaton, and gauge field, on top of this bulk configuration. The corresponding holographic dual 4 + 1 dimensional field theory in the asymptotic region is a finite-temperature system, with two conserved U(1) charges having non-vanishing vacuum expectations. We are interested in gapless collective excitations in this system, that we determine via holographic world-sheet calculation of two-point functions of the stress-energy tensor and charge currents.
In subsection 7.2, we compactify the T 4 sub-space of the coset (3.1), which allows us to study the 1 + 1 dimensional boundary field theory. The resulting expressions that we obtain for the two-point correlation functions in momentum space interpolate between the well-known CFT 2 expressions in the IR, calculated within the AdS 3 /CFT 2 correspondence, and non-local expressions for LST in the UV.

Collective excitation modes
In section 6.6, we constructed the (anti-)holomorphic vertex operators (Ṽ ( 1 2 ) ) V ( 1 2 ) describing massless NS (right-) left-moving string states. The NS-NS states of closed type-II superstring are obtained by taking a direct product of the (anti-)holomorphic states. The corresponding vertex operator V ( 1 2 ) ⊗Ṽ ( 1 2 ) can then be split into irreducible representations of the target space symmetry group, rendering the graviton, B-field, and dilaton vertex operators, and forming the NS-NS bosonic degrees of freedom of the type-II supergravity multiplet.
These operators describe excitation on top of the metric, B-field, and dilaton tendimensional background configuration (3.100), (3.103), (3.101) that we derived in section 3.2, as well as the nine-dimensional two-charge background (5.2) derived in section 5. An excitation mode is characterized by the energy κ, and momentum p. Without loss of generality, we can align the momentum with the direction z 1 on the torus T 4 , p 1 L = p 1 R = p, setting the rest of the momenta to zero, p i L,R = 0, i = 2, 3, 4. In this section, we are going to discuss holographic interpretation of string excitations in the bulk in terms of the dual 4 + 1 dimensional field theory on the boundary. 42 Recall that the field theory dual of a charged black brane describes matter at finite density. 43 In particular, we are interested in the corresponding collective excitation modes, that we can compare with the hydrodynamic theory. These appear as poles of two-point correlation functions of the stress-energy tensor. In general, these correlation functions also include conserved charge current, and form correlation matrices. Correspondingly, the bulk holographic dual of the stress-energy tensor is given by the fluctuating metric field, which is coupled to fluctuations of the gauge fields, B-field, and dilaton. 44 For the space-like polarizations m, n = x, i, where i = 1, 2, 3, 4, along the circle U(1) x and the torus sub-space T 4 , we write down the NS-NS type-II supergravity vertex operators Performing the Kaluza-Klein reduction on the U(1) x circle, we set p L = p R = 0, and derive the gauge fields vertex operators These vertex operators can be further split into three groups, defined by the spin rotation properties in the sub-space α, β = 2, 3, 4 transverse to the direction z 1 of propagation 42 This is to be contrasted with discussion in section 6.7, where we compactify T 4 and study 1 + 1 dimensional field theory on the boundary. 43 See, e.g., [62,63] and references therein. 44 See [31,33] for analogous calculation in coset theories involving SL(2, R)/U(1) and SL(2, R)×U(1)/U(1).
Notice that due to rotation symmetry, operators from different groups do not couple to each other. Furthermore, the B-field excitations and the b gauge field mix with the graviton excitations and the a gauge field. These can be decoupled from each other into the left-moving and right-moving sectors as follows: spin-0 sound channel: spin-1 shear channel: spin-2 scalar channel: Here we have (recall that m, n = 1, 2, 3, 4, and α, β = 2, 3, 4) Non-trivial two-point correlation functions in the sound channel are then given by and Similarly, the shear channel non-vanishing correlation functions are given by and (7.10)

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In these expressions we have Ψ α while the rest of the two-point functions are given by (6.76) and its anti-holomorphic counterparts. These correlation functions were shown in section 6.6 to possess simple poles at j = m and j = −m.
Let us locate the gapless shear diffusion mode. At zero propagation velocity, for the mass M , pressure P , volume V , and viscosity η, we have (for the dimensionless ω, p, measured in units of string length) Defining entropy density s = S/V , taking into account that due to (4.24) the pressure vanishes, P = F = 0, and using expressions for the mass M = E given by (3.111), and entropy S given by (4.10), we obtain 45 (7.12) At the same time, the mass-shell condition (6.56) while the BRST constraints (6.42) impose Using results of section 6.4 and performing Wick rotation, we obtain κ = −iω. From (7.14) we can therefore express while from (7.13) we derive j = 1 2 −1 + 1 + 4j (j + 1) − k 2 s (p 2 + ω 2 ) . (7.16) As discussed above, excitation modes are located at j = m and j = −m, coming from the holomorphic and the anti-holomorphic sector respectively. At j = 0, m = 0, these give rise to the modes Using this in (7.12) we obtain Analogous systems of two decoupled liquids has been found in [31]. 45 We hope that recycling the same letter η, used before as a parameter of BRST transformation (6.12), does not cause a confusion.

Correlation functions on two-dimensional boundary
In this subsection, we will compute correlation functions of operators in momentum space of the space-time theory dual to string theory on the coset (3.1) in the pure NS-NS sector. Unlike the previous section, we will work with the setup where the sub-space T 4 has been compactified. The dual boundary theory is therefore two-dimensional.
Since the space-time theory is not a local field theory, 46 it is not immediately clear if there exist local operators on the boundary theory. To get around this issue, we restrict ourselves to correlation functions of operators in momentum space. One can, of course, Fourier-transform the momentum space correlation function to position space but the interpretation of such position space two-point correlation function is not clear.
To start with, let us review the computation of two-point correlation function of local operators in CFT 2 dual to superstring theory in AdS 3 × S 3 × T 4 . A large class of scalar observables of string theory in AdS 3 × S 3 × T 4 in the (−1, −1) picture is given by the expression [8] O (7.19) where x,x are coordinates of the space-time theory, z,z are coordinates of the worldsheet Riemann surface Σ, ϕ,φ are the world-sheet superconformal ghosts, Φ h is the usual non-normalizable vertex operator of the world-sheet theory on AdS 3 with world-sheet di- V su is the N = 1 superconformal primary of the CFT on S 3 with dimensions (∆ su ,∆ su ), and V T 4 is the N = 1 superconformal primary of the CFT on T 4 with dimensions (∆ T 4 ,∆ T 4 ).
The local operator O(x,x), given by (7.19), of the space-time theory dual to string theory in AdS 3 has dimensions (h, h). For simplicity let us assume ∆ su =∆ su and ∆ T 4 =∆ T 4 . The onshell condition of the operator (7.19) reads The vertex operators V su and V T 4 are normalized so that while the world-sheet operators Φ h (z; x) are normalized so that where the coefficient B(h) is given in (2.23). Taking the Fourier transform in x,x one recovers the correlation function in momentum space p,p, Next let us focus on computation of two-point function of operators in momentum space of the boundary theory dual to string theory in the background (3.1). As stated earlier, the boundary field theory is not conformal because the bulk space-time does not asymptote to AdS 3 . One can write the (−1, −1) picture operators in momentum space in the coset construction as The onshell condition is given by where (h p , h p ) 47 is the space-time dimension of Φ hp , and h = lim |p|→0 h p . (Additionally, the BRST constraints (6.42) need to be satisfied.) Then from (7.28) we derive where the momentum of the space-time theory is given by Notice that, as before, the Φ hp , V su and V T 4 are vertex operators of the CFT on AdS 3 , SU(2), and T 4 with the two-point functions given by (7.24) (with h replaced by h p ), and (7.22). Thus, for string theory on (3.1), the two-point function of the dual spacetime theory is given by This matches exactly with the two-point function computed in similar coset backgrounds that asymptotes to a linear-dilaton background [32,64]. As expected, in the deep IR, i.e., | s p| 1, one recovers the CFT 2 correlation function (7.25). In the deep UV, 47 One can interpret (hp, hp) as the dimension of the operator O(p) in the space-time theory dual to string theory in the coset (3.1) (see [42,43] on related discussion on the momentum dependent dimension of operators in non-local CFTs).

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i.e., | s p| 1, the two-point function (7.31), however, behaves as |p| 2 √ k s|p| which is a highly non-local behavior. The normalization of the two-point function of primary operators in a CFT (7.25) can be changed by rescaling the operators by an arbitrary analytic function of their dimensions. Physics remains invariant due to such a rescaling of the operators. In the coset theory, however, it seems that the two-point function depends on the normalization of the operators due the momentum dependence of h p . At this point, it is unclear how to fix the normalization of the operators of the space-time theory.
Let us end this section by giving a brief sketch of the general structure of the three-point correlation function of the operator (7.27) on the boundary theory in momentum space, in the un-spectrally flowed sector. Analogously to calculation of the two-point function, we determine where OPE coefficients due to SL(2, R) and SU(2) factors are given by (2.26), (2.56), and

Discussion
In this paper, we have investigated type-II superstring theory on the coset space (3.1). It is described by the superconformal gWZW model at level k, such that R = √ k s is the characteristic length scale of the target space-time. We have derived an exact (in k) tendimensional target space-time background configuration of the NS-NS fields, including the metric, B-field, and dilaton. This background interpolates between local AdS 3 × S 3 × T 4 in the IR and linear dilaton times S 3 × S 1 × T 4 in the UV. It possesses an event horizon, with non-vanishing angular velocities along two of the S 3 directions, as well as along the S 1 circle. We have calculated the corresponding temperature, entropy, energy, conserved angular momenta, and free energy. Whenever applicable, our results reproduce the known expressions for thermodynamics of (charged) black brane backgrounds obtained in the nearhorizon limit of the NS5-F1 systems, and described by coset models that can be found as particular cases of (3.1).
We have also constructed a nine-dimensional background obtained by performing the KK reduction of the ten-dimensional background on the U(1) x circle S 1 . Such a background is particularly relevant in the context of holographic description of strongly-coupled systems at finite density. This is due to the fact that KK reduction on S 1 generates non-trivial profiles of gauge fields, that furnish holographic dual of non-vanishing densities of conserved U(1) charges of the boundary field theory. In our case, the role of the boundary theory is played by LST found in the asymptotic linear-dilaton region of the bulk space-time, and characterized by non-vanishing vacuum expectations of two independent U(1) charges.

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Due to exact solvability of the corresponding gWZW model, the bulk theory enjoys an exact in k world-sheet control, going beyond the effective supergravity approximation R/ s 1. In particular, it allows one to access the regime of dual field theory characterized by a finite number of degrees of freedom, determined by k. We take advantage of this power of exact world-sheet theory to calculate two-point correlation functions of massless NS-NS vertex operators describing the graviton, B-field, and dilaton excitations on top of the constructed background. Holographically, these give us correlation matrices of the stress-energy tensor and two U(1) currents of the dual field theory. These correlation matrices manifest a rich structure of collective density excitations, that include gapless diffusion modes. We derive the corresponding dispersion relations and comment on their hydrodynamic interpretation.
We also derived the two-and three-point correlation functions of the scalars in the 1 + 1-dimensional boundary theory in momentum space. The expressions for the correlation functions take the form of CFT 2 correlation functions with the dimensions h being replaced by h p (7.29). (In fact, such a prescription seem to hold for higher-order correlation functions as well.) At low energies we recover the CFT 2 correlators, but at high energies the correlators seem to develop highly non-local behavior. This is in harmony with the fact the short-distance physics is governed by LST rather than a local fixed point. However, the correlation functions at an arbitrary point on the RG flow seem to depend on the normalization of the operators at the IR fixed point. In this paper we do not provide a recipe to fix the normalization ambiguity in studying holography in non-AdS backgrounds. It would be nice to see if the Ward identities of the theory at a generic point on the RG flow can impose constraints in the normalization. It would also be interesting to perform an LSZ reduction on the correlators and keep only contributions that give rise to physical onshell particles. This may shed some light on the normalization of the operators of the boundary theory.
One of the questions left unanswered in this paper concerns microscopic interpretation of the obtained background in terms of extended string theory objects. Particularly, we expect that the background configuration that we constructed in this paper can be found in the near-horizon limit of a certain system involving separated rotating NS5-brane in the double-scaling limit [65,66] and fundamental F1-strings with non-trivial momentum quantum numbers. We believe this question can be resolved in a manner analogous to [32] that recently provided an embedding of the (SL(2, R)×U(1))/U(1) target-space background as a near-horizon limit of a certain NS5-F1 system. We would like to have an exact map between the gauge-invariant operators of the boundary theory and the world-sheet theory and identify the operators in the chiral ring algebra of the boundary theory.
The supergravity background obtained in this paper is characterized by asymptotically linear dilaton geometry. In order to describe this background as a near-horizon limit of NS5-F1 system, it would be helpful to embed it into asymptotically flat space-time. We expect that such an embedding can be done in the spirit of [67] which considered D1-D5 system. We leave this for future work.
Our work paves the avenue for various future research. In section 7.2 we computed the correlation function of scalar operators in the momentum space. It would be an interesting to check if the same correlation functions are recoverable from the supergravity analysis.

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As stated earlier, the boundary field theory is a certain vacua of LST that flows to a CFT 2 in the IR. At a generic point on the RG flow, the boundary field theory is not local. It has been shown before [68,69] that in similar non-local setups, the Ryu-Takayanagi curve develops non-local features below a certain scale (non-locality scale proportional to s ). It would be interesting to understand the non-local features of boundary theory through the lenses of holographic entanglement entropy. In the same spirit it is worth computing the holographic complexity where we expect to observe exotic divergences reflecting the non-local features of LST. It would also be nice to set up a computation of holographic Wilson-loop to understand the potential energy between a probe quark-anti-quark pair in the boundary theory [70]. From the analytic structure of the quark-anti-quark potential energy, one may expect to understand the various phases of the theory.

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At the same time, from the WZ action (A.3) we derive Performing cyclical permutation under the trace, while taking into account anti-symmetric property of the wedge product, we can re-arrange the first term in the r.h.s. of (A.5), obtaining as a result we can subsequently re-write (A.6) as Substituting here dg = ∂g dz +∂g dz, and using that we finally re-write as In particular, for the variation determined by a single local parameter ξ(z,z) and the generators T L,R belonging to a u(1) sub-algebra of the group algebra of G, where we have defined the (anti-)holomorphic U(1) currents at the level k = 1 as Using (A.13) we derive the following gauge transformations of these currents,

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From expression (A.12) one can see that it is possible to gauge the corresponding U(1) sub-group of G parametrized by ξ. To that end, introduce auxiliary (non-dynamical) gauge fields A,Ã, that transform as under the infinitezimal gauge transformation, and define the gauged WZW action as where M is a gauge-invariant function of g to be determined below. Algebraic equations of motion for the gauge fields A,Ã that follow from the action (A.

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This can also be written as where the SU(2) ∼ S 3 manifold a 2 + b 2 + c 2 + d 2 = 1 is parametrized by a = cos θ cos γ + , b = cos θ sin γ + , c = sin θ cos γ − , d = sin θ sin γ − , (A. 27) for γ ± = α ± β. After a straightforward calculation we obtain where we denoted the target-space coordinate as X µ = (α, θ, β), and defined the corresponding background metric as For the representation (A.24) we also obtain Tr g −1 dg ∧ g −1 dg ∧ g −1 dg = 6d (cos(2θ) dα ∧ dβ) , (A. 30) and therefore the Wess-Zumino term induces the background B-field via where we denoted We conclude this section by writing down expression for the volume of the unit 3sphere. Using negative of the metric (A.29), which is achieved by selecting the level k = −1, we obtain

A.3 Effective action and superconformal WZW model
We have reviewed the classical (gauged) WZW models describing bosonic string theory with the target space given by the group manifold G. In particular, we considered examples of G = SL(2, R) and G = SU (2), and derived the corresponding classical target space background. The latter is generally specified by the metric, dilaton, and B-field. The WZW model is characterized by the level k, such that the classical action of the WZW model is given by S = k S, where the k = 1 action S is given by (A.1). Perturbative corrections to the classical theory, due to the world-sheet path integral expansion, therefore behave as 1/k. The semi-classical limit is then achieved in the limit of large k.
In terms of the target space theory, the semi-classical limit translates into the small curvature or, equivalently, the large length scale limit. Indeed, from our discussion in sections A.1, A.2, one can see that the target space metric scales as k, that sets the characteristic length scale of the semi-classical geometry to be √ k s . Similar observations can be made when one considers gauged WZW models. In particular, integrating out auxiliary gauge fields, via their equations of motion, in the classical WZW action on a coset space renders a classical sigma-model. Classical background fields, including the metric, dilaton and B-field, can be read off from the sigma-model action.
We now proceed to reviewing how classical results get modified when 1/k corrections, due to expansion of the world-sheet path integral, are taken into account. It is illustrative to follow two different perspectives, that ultimately are shown to render consistent results. The first perspective involves the (anti-) holomorphic symmetry argument and the affine algebra relations for the corresponding currents, while the second one involves calculation of the effective action for the WZW model [71][72][73].

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Let us now turn to the discussion of effective action in supersymmetric WZW models. The starting point is supersymmetric action k S susy [g, χ a ,χ a ] at level k. By performing a change of fermionic degrees of freedom, we can reformulate the theory in terms of decoupled free fermions ψ a ,ψ a , discussed above. This is a chiral transformation, that produces a nontrivial functional determinant that can be reformulated as e c G S[g] . The total action is then given by k S[g] + k S f [ψ,ψ], where S f [ψ,ψ] is the action of free two-dimensional Majorana-Weyl fermions ψ a ,ψ a at level k = 1 [72,73]. Consistently with our discussion above, we recognize the shifted level k = k − c G , that appeared in the OPE (A.49) of the bosonic sector Kac-Moody currents j a .
Since the fermions ψ a ,ψ a are free and decoupled, the rest of derivation of the effective action proceeds similarly to the bosonic WZW case. Analogously to the effective action (A.45), going from classical supersymmetric WZW action to the effective action of supersymmetric WZW model, the level k gets shifted as k → k + c G = k. In other words, starting with the action k S susy [g, χ a ,χ a ] at level k, first the fermionic determinant due to the change of variables χ,χ → ψ,ψ shifted the level k prefactor of the bosonic sector action by −c G , that subsequently was offset by the +c G shift in calculation of the effective action, yielding at the end the total effective action In the main text, whenever we write down action of (gauged) WZW model, we usually have in mind supersymmetric model, considered at level k, and write down the exact worldsheet effective action thereof. This becomes important, for instance, when we write down the action on the coset (3.1), that combines contributions from the SL(2, R) and SU (2) sectors. Due to the ±c G cancellations between the level k shifts, the effective action of supersymmetric model on (3.1) is at level k both in the SL(2, R) and SU(2) sectors. While such 1/k corrections are unimportant in the semiclassical limit k → ∞, our results are therefore in fact exact at finite k in the full quantum world-sheet theory.

A.4 Some (anti-)commutation relations
In this appendix we collect various (anti-)commutation relations satisfied by the amplitudes of the superconformal and the super-Kac-Moody currents. Specifically, we are interested in the NS sector of superstring states, and therefore all the fermionic operators are expanded in the half-integer modes.

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The free energy is determined by where µ L,R are the left-and right-moving components of the chemical potential. As pointed out in [50], the free energy of the theory (C.1) vanishes, which is in agreement with our result (4.24) for (3.1) obtained independently of the choice of the anomaly-free gauging parameters a i , b i . We can in fact match expressions for the free energy term by term. We have already matched M and T S with our results in the limit of un-gauged SU (2). Now let us match µ L Q L + µ R Q R . The chemical potential can be found from the asymptotic behavior of the gauge field components 55 where the gauge fields can be found in eq. (C.9) of [51]. Using eq. (C.9) of [51], we first determine the relation y = t √ 2 sec χ+sec(χ−ψ) between the time coordinate y, used in that equation, and the time coordinate t, featuring asymptotic behavior of the metric g tt = −1.
We then obtain µ G = sec χ − ψ 2 sin ψ 2 , µ B = cos χ tan ψ 2 − sin χ. Consequently (C.9) We then obtain Open Access. This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited. SCOAP 3 supports the goals of the International Year of Basic Sciences for Sustainable Development. 55 Again, the factor of 1/ √ 2 appears when we relax the convention of α = 2 s = 2 → s = √ 2 used in [51]. This in turn rescales the time coordinate measured in units of s by the factor of 1/ √ 2.